Elasticity of flexible and semiflexible polymers with extensible bonds in the Gibbs and Helmholtz ensembles

Forced extension of a wormlike chain in the Gibbs and Helmholtz ensembles

A semiflexible polymer can be stretched by either applying a force to it or by fixing the positions of its endpoints. The two approaches generally yield different results and correspond to experiments performed in either the Gibbs or Helmholtz statistical ensembles. Here, we derive the Helmholtz force-extension relationship for the commonly used wormlike-chain model in the strongly stretched regime. By analyzing it in comparison with the Gibbs ensemble result, we show that equivalence between the two relationships is achieved only in the long-chain thermodynamic limit.

Elasticity of a Grafted Rod-like Filament with Fluctuating Bending Stiffness

Quite often polymers exhibit different elastic behavior depending on the statistical ensemble (Gibbs vs. Helmholtz). This is an effect of strong fluctuations. In particular, two-state polymers, which locally or globally fluctuate between two classes of microstates, can exhibit strong ensemble inequivalence with negative elastic moduli (extensibility or compressibility) in the Helmholtz ensemble. Two-state polymers consisting of flexible beads and springs have been studied extensively. Recently, similar behavior was predicted in a strongly stretched wormlike chain consisting of a sequence of reversible blocks, fluctuating between two values of the bending stiffness (the so called reversible wormlike chain, rWLC). In this article, we theoretically analyse the elasticity of a grafted rod-like semiflexible filament which fluctuates between two states of bending stiffness. We consider the response to a point force at the fluctuating tip in both the Gibbs and the Helmholtz ensemble. We also calculate the entropic force exerted by the filament on a confining wall. This is done in the Helmholtz ensemble and, under certain conditions, it yields negative compressibility. We consider a two-state homopolymer and a two-block copolymer with two-state blocks. Possible physical realizations of such a system would be grafted DNA or carbon nanorods undergoing hybridization, or grafted F-actin bundles undergoing collective reversible unbinding.

Temperature dependent model for the quasi-static stick-slip process on a soft substrate

The classical Prandtl-Tomlinson model is the most famous and efficient method to describe the stick-slip phenomenon and the resulting friction between a slider and a corrugated substrate. It is widely used in all studies of frictional physics and notably in nanotribology. However, it considers a rigid or undeformable substrate and therefore is hardly applicable for investigating the physics of soft matter and in particular biophysics. For this reason, we introduce here a modified model that is capable of taking into consideration a soft or deformable substrate. It is realized by a sequence of elastically bound quadratic energy wells, which represent the corrugated substrate. We study the quasi-static behavior of the system through the equilibrium statistical mechanics. We thus determine the static friction and the deformation of the substrate as a function of temperature and substrate stiffness. The results are of interest for the study of cell motion in biophysics and for haptic and tactile systems in microtechnology.

Thermal effects on fracture and the brittle-to-ductile transition

The fracture behavior of brittle and ductile materials can be strongly influenced by thermal fluctuations, especially in micro- and nanodevices as well as in rubberlike and biological materials. However, temperature effects, in particular on the brittle-to-ductile transition, still require a deeper theoretical investigation. As a step in this direction we propose a theory, based on equilibrium statistical mechanics, able to describe the temperature-dependent brittle fracture and brittle-to-ductile transition in prototypical discrete systems consisting in a lattice with breakable elements. Concerning the brittle behavior, we obtain closed form expressions for the temperature-dependent fracture stress and strain, representing a generalized Griffith criterion, ultimately describing the fracture as a genuine phase transition. With regard to the brittle-to-ductile transition, we obtain a complex critical scenario characterized by a threshold temperature between the two fracture regimes (brittle and ductile), an upper and a lower yield strength, and a critical temperature corresponding to the complete breakdown. To show the effectiveness of the proposed models in describing thermal fracture behaviors at small scales, we successfully compare our theoretical results with molecular dynamics simulations of Si and GaN nanowires.

Elastic traits of the extensible discrete wormlike chain model

Polymer models play the special role of elucidating the elementary features describing the physics of long molecules and become essential to interpret the measurements of their magnitudes. In this work the end-to-end distance of an extensible discrete wormlike chain polymer as a function of the applied force has been calculated both numerically and analytically, the latter as an effective approximation. The numerical evaluation uses the transfer matrix formalism to obtain an exact calculation of the partition function, while the analytic derivations generalize the simple phenomenological formulas largely used up to now. The obtained formulas are simple enough to be implemented in the fit analysis of experimental data of semiflexible extensible polymers, with the result that the elastic parameters obtained are compatible with previous measurements, and more, their accuracy strongly improves in a large range of chain extensibility.

Compression-induced buckling of a semiflexible filament in two and three dimensions

The ability of biomolecules to exert forces on their surroundings or resist compression from the environment is essential in a variety of biologically relevant contexts. For filaments in the low-temperature limit and under a constant compressive force, Euler buckling theory predicts a sudden transition from a compressed to a bent state in these slender rods. In this paper, we use a mean-field theory to show that if a semiflexible chain is compressed at a finite temperature with a fixed end-to-end distance (permitting fluctuations in the compressive forces), it exhibits a continuous phase transition to a buckled state at a critical level of compression. We determine a quantitatively accurate prediction of the transverse position distribution function of the midpoint of the chain that indicates this transition. We find the mean compressive forces are non-monotonic as the extension of the filament varies, consistent with the observation that strongly buckled filaments are less able to bear an external load. We also find that for the fixed extension (isometric) ensemble, the buckling transition does not coincide with the local minimum of the mean force (in contrast to Euler buckling). We also show the theory is highly sensitive to fluctuations in length in two dimensions, and that the buckling transition can still be accurately recovered by accounting for those fluctuations. These predictions may be useful in understanding the behavior of filamentous biomolecules compressed by fluctuating forces, relevant in a variety of biological contexts.

Freely jointed chain models with extensible links

Analytical relations for the mechanical response of single polymer chains are valuable for modeling purposes, on both the molecular and the continuum scale. These relations can be obtained using statistical thermodynamics and an idealized single-chain model, such as the freely jointed chain model. To include bond stretching, the rigid links in the freely jointed chain model can be made extensible, but this almost always renders the model analytically intractable. Here, an asymptotically correct statistical thermodynamic theory is used to develop analytic approximations for the single-chain mechanical response of this model. The accuracy of these approximations is demonstrated using several link potential energy functions. This approach can be applied to other single-chain models, and to molecular stretching in general.

Nonequilibrium free energy during polymer chain growth

During fast diffusion-influenced polymerization, nonequilibrium behavior of the polymer chains and the surrounding reactive monomers has been reported recently. Based on the laws of thermodynamics, the emerging nonequilibrium structures should be characterizable by some “extra free energy” (excess over the equilibrium Helmholtz free energy). Here, we study the nonequilibrium thermodynamics of chain-growth polymerization of ideal chains in a dispersion of free reactive monomers, using off-lattice, reactive Brownian dynamics computer simulations in conjunction with approximative statistical mechanics and relative entropy (Gibbs–Shannon and Kullback–Leibler) concepts. In the case of fast growing polymers, we indeed report increased nonequilibrium free energies Δ Fneq of several kB T compared to equilibrium and near-equilibrium, slowly growing chains. Interestingly, Δ Fneq is a non-monotonic function of the degree of polymerization and thus also of time. Our decomposition of the thermodynamic contributions shows that the initial dominant extra free energy is stored in the nonequilibrium inhomogeneous density profiles of the free monomer gas (showing density depletion and wakes) in the vicinity of the active center at the propagating polymer end. At later stages of the polymerization process, we report significant extra contributions stored in the nonequilibrium polymer conformations. Finally, our study implies a nontrivial relaxation kinetics and “restoring” of the extra free energy during the equilibration process after polymerization.

Brownian lithographic polymers of steric lock-and-key colloidal linkages

We design and lithographically fabricate two-dimensional preassembled colloidal linkages of custom-shaped, discrete, mobile microscale tiles that are sterically coupled together by lock-and-key sub-tile features, yielding hinge-like bonds between separate tiles. These mobile colloidal linkages, which we call polylithomers, provide top-down, preconfigured, morphologically controllable analogs of fluctuating molecular polymers. We illustrate the versatility of this approach by fabricating and studying curvilinear, branched, bridged-spiral, dendritic, and mesh-like polylithomers having controllable preassembled dimensions, topologies, configurations, intrinsic local curvatures, persistence lengths, and bond extensibilities. By advancing anisotropic particle tracking routines to handle lock-and-key tiles, we measure the dynamic conformational changes of polylithomers caused by Brownian excitations to the monomer scale, revealing markedly large bond extensibilities. Beyond modeling fluctuating semiflexible molecular polymers, polylithomers provide access to unusual polymer morphologies and bonding potentials that have not yet been synthesized through other kinds of assembly methods using either molecular or colloidal monomers.

Tensile elasticity of a freely jointed chain with reversible hinges

Many biopolymers exhibit reversible conformational transitions within the chain, which affect their bending stiffness and their response to a stretching force. For example, double stranded DNA may have denatured "bubbles" of unzipped single strands which open and close randomly. In other polymers, the transitions may be due to the reversible attachment and detachment of ligands on ligand-receptor complexes along the backbone. Semiflexible bundles under tension formed by the reversible attachment of cross-linkers, on a coarse-grained level, exhibit similar behaviour. The simplest theoretical model which captures what the above mentioned systems have in common is a freely jointed chain (FJC) with reversible hinges. Each hinge can be open, as in the usual FJC, or closed forcing the adjacent segments to align (stretch). In this article, we analyse it in the Gibbs ensemble. Remarkably, even though the usual FJC in the thermodynamic limit exhibits ensemble equivalence, the reversible FJC exhibits ensemble inequivalence. Even though a mean field treatment suggests a continuous phase transition to a fully hinged state at a certain force, the generating function method ("necklace model") shows that there is no phase transition. However, there is a crossover between the two states with clearly different responses. In the low force (linear response) regime, the reversible FJC has higher tensile compliance than its usual counterpart. In contrast, in the strong force regime, the tensile compliance of the reversible FJC is much lower than that of the usual FJC.

Rate-dependent force-extension models for single-molecule force spectroscopy experiments

Single-molecule force spectroscopy techniques allow for the measurement of several static and dynamic features of macromolecules of biological origin. In particular, atomic force microscopy, used with a variable pulling rate, provides valuable information on the folding/unfolding dynamics of proteins. We propose here two different models able to describe the out-of-equilibrium statistical mechanics of a chain composed of bistable units. These latter represent the protein domains, which can be either folded or unfolded. Both models are based on the Langevin approach and their implementation allows for investigating the effect of the pulling rate and of the device intrinsic elasticity on the chain unfolding response. The theoretical results (both analytical and numerical) have been compared with experimental data concerning the unfolding of the titin and filamin proteins, eventually obtaining a good agreement over a large range of the pulling rates.

Statistical mechanical constitutive theory of polymer networks: The inextricable links between distribution, behavior, and ensemble

A fundamental theory is presented for the mechanical response of polymer networks undergoing large deformation which seamlessly integrates statistical mechanical principles with macroscopic thermodynamic constitutive theory. Our formulation permits the consideration of arbitrary polymer chain behaviors when interactions among chains may be neglected. This careful treatment highlights the naturally occurring correspondence between single-chain mechanical behavior and the equilibrium distribution of chains in the network, as well as the correspondences between different single-chain thermodynamic ensembles. We demonstrate these important distinctions with the extensible freely jointed chain model. This statistical mechanical theory is then extended to the continuum scale, where we utilize traditional macroscopic constitutive theory to ultimately retrieve the Cauchy stress in terms of the deformation and polymer network statistics. Once again using the extensible freely jointed chain model, we illustrate the importance of the naturally occurring statistical correspondences through their effects on the stress-stretch response of the network. We additionally show that these differences vanish when the number of links in the chain becomes sufficiently large enough, and discuss why certain methods perform better than others before this limit is reached.

The influence of device handles in single-molecule experiments

We deduce a fully analytical model to predict the artifacts of the device handles in single molecule force spectroscopy experiments. As we show, neglecting the handle stiffness can lead to crucial overestimation or underestimation of the stability properties and unfolding thresholds of multistable molecules.

Unveiling the influence of device stiffness in single macromolecule unfolding

Single-molecule stretching experiments on DNA, RNA, and other biological macromolecules opened up the possibility of an impressive progress in many fields of life and medical sciences. The reliability of such experiments may be crucially limited by the possibility of determining the influence of the apparatus on the experimental outputs. Here we deduce a model that let us analytically evaluate such influence, fundamental for the interpretation of Single Molecule Force Spectroscopy experiments and intermolecular interactions phenomena. As we show, our model is coherent with previous numerical results and quantitively reproduce AFM experimental tests on titin macromolecules and P-selectin with variable probe stiffnesses.

Full Statistics of Conjugated Thermodynamic Ensembles in Chains of Bistable Units

The statistical mechanics and the thermodynamics of small systems are characterized by the non-equivalence of the statistical ensembles. When concerning a polymer chain or an arbitrary chain of independent units, this concept leads to different force-extension responses for the isotensional (Gibbs) and the isometric (Helmholtz) thermodynamic ensembles for a limited number of units (far from the thermodynamic limit). While the average force-extension response has been largely investigated in both Gibbs and Helmholtz ensembles, the full statistical characterization of this thermo-mechanical behavior has not been approached by evaluating the corresponding probability densities. Therefore, we elaborate in this paper a technique for obtaining the probability density of the extension when force is applied (Gibbs ensemble) and the probability density of the force when the extension is prescribed (Helmholtz ensemble). This methodology, here developed at thermodynamic equilibrium, is applied to a specific chain composed of units characterized by a bistable potential energy, which is able to mimic the folding and unfolding of several macromolecules of biological origin.

Spin variable approach for the statistical mechanics of folding and unfolding chains

The force-extension response of chains composed of bistable (or multistable) units strongly depends on the applied boundary conditions. As a matter of fact, isotensional conditions (soft devices) lead to a plateau-like response, whereas isometric conditions (hard devices) lead to a sawtooth-like pattern. We develop an equilibrium statistical mechanics methodology, based on the introduction of a set of discrete or spin variables, which is able to describe the thermal and mechanical properties of a folding and unfolding chain under arbitrary external conditions. In particular, we will work within the Gibbs and Helmholtz ensembles, which correspond to soft and hard devices, respectively. We introduce a one-dimensional system composed of multistable units and a bistable freely jointed chain. For both systems we obtain explicit expressions for the force-extension relation and we study the spinoidal behavior induced by the isometric conditions.

Stochastic mechanical degradation of multi-cracked fiber bundles with elastic and viscous interactions

The mechanics of fiber bundles has been largely investigated in order to understand their complex failure modes. Under a mechanical load, the fibers fail progressively while the load is redistributed among the unbroken fibers. The classical fiber bundle model captures the most important features of this rupture process. On the other hand, the homogenization techniques are able to evaluate the stiffness degradation of bulk solids with a given population of cracks. However, these approaches are inadequate to determine the effective response of a degraded bundle where breaks are induced by non-mechanical actions. Here, we propose a method to analyze the behavior of a fiber bundle, undergoing a random distribution of breaks, by considering the intrinsic response of the fibers and the visco-elastic interactions among them. We obtain analytical solutions for simple configurations, while the most general cases are studied by Monte Carlo simulations. We find that the degradation of the effective bundle stiffness can be described by two scaling regimes: a first exponential regime for a low density of breaks, followed by a power-law regime at increasingly higher break density. For both regimes, we find analytical effective expressions described by specific scaling exponents.

Energy dissipation and recovery in a simple model with reversible cross-links

Reversible cross-linking is a method of enhancing the mechanical properties of polymeric materials. The inspiration for this kind of cross-linking comes from nature, which uses this strategy in a large variety of biological materials to dramatically increase their toughness. Recently, first attempts were made to transfer this principle to technological applications. In this study, Monte Carlo simulations are used to investigate the effect of the number and the topology of reversible cross-links on the mechanical performance of a simple model system. Computational cyclic loading tests are performed, and the work to fracture and the energy dissipation per cycle are determined, which both increase when the density of cross-links is increased. Furthermore, a different topology of the bonds may increase the work to fracture by a factor of more than 2 for the same density. This dependence of the mechanical properties on the topology of the bonds has important implications on the self-healing properties of such systems, because only a fast return of the system to its unloaded state after release of the load ensures that the optimal topology may form.

Structure and mechanical characterization of DNA i-motif nanowires by molecular dynamics simulation

We studied the structure and mechanical properties of DNA i-motif nanowires by means of molecular dynamics computer simulations. We built up to 230 nm-long nanowires, based on a repeated TC5 sequence from crystallographic data, fully relaxed and equilibrated in water. The unusual C⋅C(+) stacked structure, formed by four ssDNA strands arranged in an intercalated tetramer, is here fully characterized both statically and dynamically. By applying stretching, compression, and bending deformations with the steered molecular dynamics and umbrella sampling methods, we extract the apparent Young's and bending moduli of the nanowire, as well as estimates for the tensile strength and persistence length. According to our results, the i-motif nanowire shares similarities with structural proteins, as far as its tensile stiffness, but is closer to nucleic acids and flexible proteins, as far as its bending rigidity is concerned. Furthermore, thanks to its very thin cross section, the apparent tensile toughness is close to that of a metal. Besides their yet to be clarified biological significance, i-motif nanowires may qualify as interesting candidates for nanotechnology templates, due to such outstanding mechanical properties.